We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $α\in (1,2]$. We prove that when conditioning such maps to have $n$ vertices, or $n$ edges, or $n$ faces, the vertex-set endowed with the graph distance suitably rescaled converges in distribution towards the celebrated Brownian map when $α=2$, and, after extraction of a subsequence, towards another `$α$-stable map' when $α<2$, which improves on a first result due to Le Gall & Miermont who assumed slightly more regularity.